# Algorithms for Sparse Random 3XOR: The Low-Density Case

Abstract : We present algorithms for variants of the 3XOR problem with lists consisting of random sparse $n$-bit vectors. We consider two notions of sparsity: low-density (each bit is independently biased towards zero) and low-weight (the Hamming weight of $n$-bit vectors is fixed). We show that the random sparse 3XOR problem can be solved in strongly subquadratic time, namely less than $\bigO{N^{2-\epsilon}}$ operations for a constant $\epsilon > 0$. This stands in strong contrast with the regular case, where it has not been possible to have the exponent drop below $2 - o(1)$. In the low-density setting, a very simple algorithm even runs in linear time with overwhelming success probability when the density is less than $0.0957$. Our algorithms exploit the randomness (and sparsity) of the input in an essential way.
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https://hal.archives-ouvertes.fr/hal-02306917
Contributor : Charles Bouillaguet Connect in order to contact the contributor
Submitted on : Saturday, October 2, 2021 - 5:57:12 PM
Last modification on : Friday, October 8, 2021 - 4:28:07 PM

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• HAL Id : hal-02306917, version 4

### Citation

Charles Bouillaguet, Claire Delaplace, Antoine Joux. Algorithms for Sparse Random 3XOR: The Low-Density Case. 2021. ⟨hal-02306917v4⟩

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